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<h3 class="heading"><span class="type">Paragraph</span></h3>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\frac{\textrm{d} v}{\textrm{d} x}=C_2 \frac{x}{(\sin x^2)^2}~\to~v=C_2 \int \frac{x}{(\sin x^2)^2} \textrm{d} x+C_3.
\end{equation*}
</div>
<p class="continuation">Letting <span class="process-math">\(t=x^2\text{,}\)</span> one has <span class="process-math">\(\textrm{d} t=2x \textrm{d} x\)</span> and the above solution to <span class="process-math">\(v\)</span> becomes</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
v=C_2 \int \frac{\frac{1}{2}}{(\sin t)^2}    \textrm{d}t+C_3=-\frac{C_2}{2} \cot t+C_3=-\frac{C_2}{2} \cot x^2+C_3.
\end{equation*}
</div>
<p class="continuation">Choose <span class="process-math">\(C_2=-2, C_3=0\text{,}\)</span> we have the second solution to be</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y_2=v y_1=\cot x^2 \sin x^2=\cos x^2.
\end{equation*}
</div>
<p class="continuation">Check</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
W(y_1, y_2)=\left|
\begin{array}{ll}
y_1 &amp; y_2\\
y_1^{\prime} &amp; y_2^{\prime}
\end{array}
\right|
=
\left|
\begin{array}{ll}
\sin x^2 &amp; \cos x^2\\
2 x \cos x^2 &amp; -2 x \sin x^2
\end{array}
\right|=-2 x \neq 0.
\end{equation*}
</div>
<p class="continuation">Thus, the general solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y=C_1 \sin x^2+C_2 \cos x^2.
\end{equation*}
</div>
<span class="incontext"><a href="sec3_5.html#p-112" class="internal">in-context</a></span>
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